Kostka polynomials and affine Kac-Moody algebras

PI建议研究仿射(以及更一般的可对称化)Kac-Moody代数的Hall-Littlewood和Kostka-Foulkes多项式的某些推广的表示理论和组合。具体地说，PI希望从Cherednik-Macdonald理论、q-超几何级数和正性的角度来研究仿射Kostka-Foulkes多项式。PI还希望使用对应于任意Kac-Moody代数的多项式来更好地阐明这些鲜为人知的李代数的表示。对称函数是当变量被置换时保持不变的多元多项式。他们承认了一个涉及代数、组合学和表示论相互作用的丰富理论。Hall-Littlewood多项式是依赖于一个额外参数的对称函数，它在两类非常重要的对称函数之间进行插补。霍尔-利特尔伍德多项式的理论可以追溯到霍尔、利特尔伍德以及后来的麦克唐纳的作品。这些多项式自然而然地出现在数学的各个领域，包括代数、几何、表示论、组合学和数学物理。它们具有许多重要的性质和应用。本研究的目的是研究经典的Hall-Littlewood多项式在无限维(Kac-Moody)李代数中的推广，并在这种一般情况下得到类似的经典性质。PI希望一方面澄清Hall-Littlewood多项式的推广与它所涉及的经典对象(如“双仿射Hecke代数”)的推广之间的关系。

The PI proposes to study the representation theory and combinatoricsunderlying certain generalizations of Hall-Littlewood and Kostka-Foulkes polynomials for affine (and more generally symmetrizable) Kac-Moody algebras. Specifically, the PI wishes to study affine Kostka-Foulkes polynomials from the perspectives of Cherednik-Macdonald theory, q-hypergeometric series and positivity. The PI also hopes to use the polynomials corresponding to arbitrary Kac-Moody algebras toshed more light on representations of these poorly understood Lie algebras.Symmetric functions are multivariate polynomials which remain unchangedwhen the variables are permuted. They admit a rich theory involving an interplay of algebra, combinatorics and representation theory. Hall-Littlewood polynomials are symmetric functions that depend on an extra parameter and interpolate between two very important classes of symmetric functions. The theory of Hall-Littlewood polynomials traces its origins to works of Hall, Littlewood and later of Macdonald. These polynomials occur naturally in diverse areas of mathematics includingalgebra, geometry, representation theory, combinatorics and mathematical physics. They have many important properties and applications.The goal of the proposed research is to study the generalization of the classical Hall-Littlewood polynomials to the case of infinite dimensional (Kac-Moody) Lie algebras and to derive analogs of classical properties in this general setting. The PI hopes to clarify the relation between the generalizations of Hall-Littlewood polynomials on one hand and the generalizations of the classical objects it is related to (such as the "double affine Hecke algebra").